The Spectral Geometry on the submanifold of (pseudo-) Euclidean space

我们在这个项目中获得了25张或更多的照片(见参考文献)。在19971。得到了伪欧氏空间中双调和子流形分类的一些新结果。具体地，(1)完成了伪欧氏空间中双调和曲线的分类问题。(2)证明了三维伪欧氏空间中不存在双调和曲面。(3)得到了四维伪欧氏空间中双调和曲面的分类定理。对于伪欧氏空间中具有Ricci曲率的类空子流形，讨论了它们的分类。对于伪欧氏空间中De-Sitter球面中的常数量曲率超曲面，讨论了球面定理。对于共形平坦的三维黎曼流形，在一定的Ricci曲率和标量曲率条件下，讨论了它们的分类问题。19981。…进一步，在标量曲率的条件下，我们得到了伪欧氏空间中任何浸入de-Sitter球面的紧致子流形只是一个标准空间。得到了Clifford环面的一个刻画。(1)讨论了具有非负常数量曲率的三维共形平坦黎曼流形和Ricci曲率的常范数。(2)讨论了三维共形平坦的负常数量曲率的黎曼流形和Ricci曲率的常范数。讨论了具有Ricci曲率有界范数的单位球面上极小闭曲面的分类问题。这一结果与S.S.Chern，Do Carmo和S.Kobayashi的著名定理一致，即Clifford环面只是单位空间中S=n的极小闭曲面。19983年度第三次上市工作取得了一定进展。我们现在研究由Bang-yen Chen提出的下列公开问题：(1)三维欧氏空间中的有限型曲面的分类问题。(2)n维欧氏空间中的双调和子流形的分类问题。(3)四维Minkovski空间中的双调和子流形的分类问题。较少

We obtained 25 and more prints in this Project (See REFERENCES).In 19971. We obtained some new results about the classification of biharmonic submanifold in psudo-Euclidean space. In detail,(1) The classification problem of biharmonic curves in psudo-Euclidean space was completed.(2) It was proved that the bihaemonic surfaces do not exist in 3 dimensional psudo-Euclidean space.(3) Some classification theorems of biharmonic surfaces in 4 dimensional psudo-Euclidean space was obtained.2. About the spacelike maximal submanifolds with some conditions for Ricci curvature immersed in de-Sitter sphere in psudo-Euclidean space, the classification of them was discussed.3. About the hypersurfaces of constant scalar curvature immersed in de-Sitter sphere in psudo-Euclidean space, the sphere theorem was discussed.4. About the comformally flat 3 dimensional Riemannian manifolds under some conditions for Ricci curvature and scalar curvature, the classification problem of them was discussed.In 19981. … More We obtained, under some conditions for scalar curvature, that any compact submanifold immersed in de-Sitter sphere in psudo-Euclidean space is only a standard sphere.2. We obtained a characterization about the Clifford torus.3. (1) We discussed about 3-dimensional comformally flat Riemannian manifold with non negative constant scalar curvature and the constan norm of Ricci curvature.(2) We discussed about 3-dimensional comformally flat Riemnnian manifold with negative constant scalar curvature and the constan norm of Ricci curvature.In 19991. We discussed the classification problem about the minimal closed surfaces in unit sphere with bounded norm of Ricci curvature. This result is concerted with the famous theorem by S.S.Chern, do Carmo and S. Kobayashi that the Clifford torus is only minimal closed surfaces of S=n in unit sphere.2. We obtained some progress concerned with the third work of listed in 19983. We now investigate the following open problems proposed by Bang-yen Chen;(1) The classification problem of the finite type surface in 3 Euclidean space.(2) The classification problem of the biharmonic submanifolds in n-dimensional Euclidean space.(3) The classification problem of the biharmonic submanifolds in 4-dimensional Minkovski space. Less